Simplify and expand the following expression: $ \dfrac{5}{5x + 25}- \dfrac{1}{3x + 27}+ \dfrac{2x}{x^2 + 14x + 45} $
First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $5$ out of denominator in the first term: $ \dfrac{5}{5x + 25} = \dfrac{5}{5(x + 5)}$ We can factor a $3$ out of denominator in the second term: $ \dfrac{1}{3x + 27} = \dfrac{1}{3(x + 9)}$ We can factor the quadratic in the third term: $ \dfrac{2x}{x^2 + 14x + 45} = \dfrac{2x}{(x + 5)(x + 9)}$ Now we have: $ \dfrac{5}{5(x + 5)}- \dfrac{1}{3(x + 9)}+ \dfrac{2x}{(x + 5)(x + 9)} $ The least common multiple of the denominators is: $ 15(x + 5)(x + 9)$ In order to get the first term over $15(x + 5)(x + 9)$ , multiply by $\dfrac{3(x + 9)}{3(x + 9)}$ $ \dfrac{5}{5(x + 5)} \times \dfrac{3(x + 9)}{3(x + 9)} = \dfrac{15(x + 9)}{15(x + 5)(x + 9)} $ In order to get the second term over $15(x + 5)(x + 9)$ , multiply by $\dfrac{5(x + 5)}{5(x + 5)}$ $ \dfrac{1}{3(x + 9)} \times \dfrac{5(x + 5)}{5(x + 5)} = \dfrac{5(x + 5)}{15(x + 5)(x + 9)} $ In order to get the third term over $15(x + 5)(x + 9)$ , multiply by $\dfrac{15}{15}$ $ \dfrac{2x}{(x + 5)(x + 9)} \times \dfrac{15}{15} = \dfrac{30x}{15(x + 5)(x + 9)} $ Now we have: $ \dfrac{15(x + 9)}{15(x + 5)(x + 9)} - \dfrac{5(x + 5)}{15(x + 5)(x + 9)} + \dfrac{30x}{15(x + 5)(x + 9)} $ $ = \dfrac{ 15(x + 9) - 5(x + 5) + 30x} {15(x + 5)(x + 9)} $ Expand: $ = \dfrac{15x + 135 - 5x - 25 + 30x}{15x^2 + 210x + 675} $ $ = \dfrac{40x + 110}{15x^2 + 210x + 675}$ Simplify: $ = \dfrac{8x + 22}{3x^2 + 42x + 135}$